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College Physics

Reduced density matrix and its applications in quantum information processing

Hong‐Yi Dai

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The density matrix expands quantum state in quantum machine.The reduced density matrix describing the system of interest is obtained from the total density matrix by taking the trace over all variables of the unobserved system.The reduced density matrix is introduced,its applications in quantum information processing such as the definition of an entangled state,entanglement quantification and quantum decoherence are pointed out.

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The density matrix expands quantum state in quantum machine.The reduced density matrix describing the system of interest is obtained from the total density matrix by taking the trace over all variables of the unobserved system.The reduced density matrix is introduced,its applications in quantum information processing such as the definition of an entangled state,entanglement quantification and quantum decoherence are pointed out.
Density matrix is one important tool in quantum mechanics,and it has very broad applications.However there are different definitions about the density matrix,and they describe quite different systems.There has been some misunderstanding about the density matrix in the community,and these misunderstandings hinder the right application of the density matrix.In this article,we discuss the different definitions of density matrix. We suggest to use the full density matrix,compressed density matrix and the reduced density matrix to describe the state of a complete quantum system,the state of an averaged particle in an ensemble and the state of part of a composite system.We stress that contrary to the wide accepted understanding that ensembles with the same compressed density matrix are physically indistinguishable, they are distinguishable through the so-called ensemble measurement.As an application,we suggest that the present conclusion that the present-day nuclear magnetic resonance quantum computation does not have quantum entanglement is groundless.
A new quantum mechanical notion - Conditional Density Matrix - proposed by the authors, is discussed and is applied to describe some physical processes. This notion is a natural generalization of von Neumann density matrix for such processes as divisions of quantum systems into subsystems and reunifications of subsystems into new joint systems. Conditional Density Matrix assigns a quantum state to a subsystem of a composite system under condition that another part of the composition system is in some pure state. It is shown that conditional density matrix naturally arises by expanding of reduced density matrix.
We explore the main processes involved in the evolution of general quantum systems by means of Diagrams of States, a novel method to graphically represent and analyze how quantum information is elaborated during computations performed by quantum circuits. We present quantum diagrams of states for representations of quantum states by density matrices, partial trace operations, density matrix purification and time-evolution by Kraus operators. Following these representations, we describe by diagrams of states the most general transformations related to singlequbit decoherence and errors. Diagrams of states prove to be a useful approach to analyze quantum computations, by offering an intuitive graphic representation of the processing of quantum information. They also help in conceiving novel quantum computations, from describing the desired information processing to deriving the final implementation by quantum gate arrays.
We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in contrast to a general belief that ensembles with the same compressed density matrix are identical. Explicit expression for the fluctuation of an observable in a specified ensemble is given. We have discussed the nature of nuclear magnetic resonance quantum computing. We show that the conclusion that there is no quantum entanglement in the current nuclear magnetic resonance quantum computing experiment is based on the unjustified belief that ensembles having the same compressed density matrix are identical physically. Related issues in quantum communication are also discussed.
A quantum system is always in interaction with its environment. This interaction inevitably alters the state of the quantum system, which causes loss of information encoded in the system. The system under consideration is not a closed system any more when interaction with the outside world is in action. We formulate the theory of an open quantum system in this chapter by regarding the combined system of the quantum system and its environment as a closed system and subsequently trace out the environmental degrees of freedom. Let ρS and ρE be the density matrices of the system and the environment, respectively. Even when the initial state is an uncorrelated state ρS ⊗ ρE , the system-environment interaction entangles the total system so that the total state develops to an inseparable entangled state in general. Decoherence is a process in which environment causes various changes in the quantum system, which manifests itself as undesirable noise.
CConsider a bipartite quantum system consisting of two subsystems A and B. The reduced density matrix ofA a is obtained by taking the partial trace with respect to B. In this work, we will show that the Wigner distribution of this reduced density matrix is obtained by integrating the total Wigner distribution with respect to the phase space variables corresponding to subsystem B. The proof we give is rigorous (as opposed to those found in the literature) and makes use of the Weyl--Wigner--Moyal phase space formalism. Our main result is applied to general Gaussian mixed states, of which it gives a particularly simple and precise description. We also briefly discuss the purification of a mixed state from the Wigner formalism point of view.